We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equals 2) and our second family is given as Cayley graphs of finite groups with very short presentations with only 2 generators and 4 relations. Both families are based on particular finite q...

We completely characterize real Bott manifolds up to diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with Z/2 coefficients is induced by an affine diffeomorphism betwe...

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra LG of countable directed graph G as well as its norm-closed counterpart, the tensor algebra T (G). We prove that the finite dimensional nest representations separate the points in LG, and a fortiori, in T (G). The irreducible finite dimensional representations separate the points in LG if and only ...

2–ENUMERATIONS OF HALVED ALTERNATING SIGN MATRICES THERESIA EISENKÖLBL Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail: [email protected] Abstract. We compute 2–enumerations of certain halved alternating sign matrices. In one case the enumeration equals the number of perfect matchings of a halved Aztec diamond. In the other case the e...

We study drawings of graphs of maximum degree six on the hexagonal (triangular) grid, with the main focus of keeping the number of bends small. We give algorithms that achieve 3.5n + 3.5 bends for all simple graphs. We also prove optimal lower bounds on the number of bends for K7, and give asymptotic lower bounds for graph classes of varying connectivity.

This paper introduces a special graph family, together with an extensive characterization of some of its properties and two of its immediate applications. The graph denoted by Ti,j, is a regular hexavalent toroidal graph. The topological features of Ti,j include vertex symmetry, Hamiltonian decomposition, translation and rotation isomorphism. Topologically, the graph is a torus, while algebraic...

A colouring of a strongly regular graph is an allocation of colours (or treatments) to the vertices of the graph. Such a colouring is balanced if every pair of distinct colours occurs equally often on the ends of an edge. When the graph is the complete regular multipartite graph a balanced colouring is just a balanced incomplete-block design, or 2-design. Some constructions are given. For examp...

We prove a spectral theorem for bimodules in the context of graph C∗-algebras. A bimodule over a suitable abelian algebra is determined by its spectrum (i.e., its groupoid partial order) iff it is generated by the Cuntz– Krieger partial isometries which it contains iff it is invariant under the gauge automorphisms. We study 1-cocycles on the Cuntz–Krieger groupoid associated with a graph C∗-alg...

In this paper, we describe a method for refining a class of balanced bintree triangulations which maintains a hamiltonian cycle in the dual graph. We also introduce a method for building refinable balanced bintree triangulations using two types of tiles, a diamond tile and a triangular tile.